if you have already established that V = {a0 + a1x : a0,a1 in F} (you don't SAY what F is, but presumably it is the field of real numbers, it doesn't HAVE to be, though...there ARE other fields) is a vector space, you don't have to "re-prove" commutativity of vector addition or associativity of vector addition, since these things hold for ALL elements of V (including those that happen to be in P1). we say such properties are "inherited" by any subset.

you WILL have to show that P1 has a 0-vector (or at the very least that at least ONE polynomial of degree 1 or less, or the 0-polynomial, satisfies your membership requirement for P1).

if you show "closure under scalar multiplication", then the two distributive laws, and the unit scalar multiplication law are also "inherited" properties.

Re: vector space

oh ok thanks. I have a quick question. Since P1={a0 + a1x|a0 - 2a1 = 0} where a0 - 2a1 = 0. Also a0=2a1. Could I rewrite the set as such P1={2a1 + a1x|a0=2a1). I still get the same elements either way.